Stability Analysis of Quadrature Methods for Two-Dimensional Singular Integral Equations
- Abdel-Fattah, Ibrahim Saad
2010 Mathematics Subject Classification
- 45L10 45Exx 65N38 65R20
- singular integral equation, two-dimensional manifold, quadrature method
In this paper we apply a quadrature method based on the tensor product trapezoidal rule to the solution of a singular integral equation over the two-dimensional torus. We prove that this method is stable if and only if a certain numerical symbol does not vanish. For a special kernel function, we present a plot of numerically computed symbol values and, for symmetric kernels (Mikhlin-Giraud kernels), we show that the symbol is different from zero if the singular integral operator is invertible. Finally, we prove the convergence of our method and present numerical tests.